Section 1 The Good Problems Method
The main goal of this method is to teach mathematical writing. It was motivated by our experience grading technology-based laboratory reports in sophomore-level courses. It seemed that most of them had never before been asked to write a coherent report on a mathematical subject. They had not mastered even a basic set of writing skills. Furthermore, they resented being asked to write, because they believed that it was not necessary for engineers and scientists to know how to write. For most college courses this is true, but we know that after college, it is crucial to be able to write about what you do.
The method itself is very simple. Each week, in conjunction with the regular homework assignment, one homework problem will be designated a “good problem.” The student must write the solution to this problem carefully, demonstrating specific skills at presenting mathematics. A set of six skills will be taught during the course of the semester or quarter. At any given point in the term, the student will be graded only on those skills that have already been taught.
The main mechanism for teaching these skills is a set of skill pages, which were 2-page handouts in the original print version. Each “handout” identifies and explains a specific skill, and provides examples of good use of this skill and examples where this skill is lacking. The examples are taken from differential Calculus, so the handouts can be used at that level or above. A secondary mechanism for teaching these skills can be discussions or demonstrations during the class or recitation. After a skill is officially learned, the regular feedback from these weekly assignments will coax the students toward mastery of that skill.
One beneficial side effect of this program is to encourage organization. College presents new-found pressures on the students’ time and energy. It is expected that to survive the student will have to learn time-management and organizational skills. This topic is beyond the scope of our endeavor, but we require the student to present one well-organized product each week.
A second beneficial side effect is to encourage logical thinking. This is another skill that is vital, but not taught directly. By teaching the students the proper uses of logical connectives, we can teach them to recognize when their argument has gaps or contradictions. It is again beyond our scope to teach logical thinking properly, but we can require one logical product each week, and provide some basic skills.
One very important aspect of this method is that it is designed to be as painless as possible. One drawback of many teaching reforms is that they require significantly more effort on the part of both the instructors and the students. When performance gains are noted, it is not clear how much is simply due to spending more time thinking about mathematics. We aim here not to add effort to the system, but to re-align a small amount of effort to a more productive area.
This documnent contains all the materials required to implement this method. There is some initial added labor involved of course, but then instructors need merely assemble the ingredients provided and set the program in motion. The instructors or teaching assistants will need to spend time brushing up on their own writing skills, but this will benefit them, and so should not be considered a burden. In the short term, we are requiring the students to learn additional skills and they will therefore have to expend additional effort. We expect, however, the returns to be rather quick. Many of these ‘presentation’ skills transfer directly to problem-solving skills, and also help with reading mathematics. We expect these skills to carry over to future mathematics courses.
